7.6 The Hyperbolic Functions Return To Contents Go To Problems & Solutions The hyperbolic functions are defined in terms of the natural exponential function ex. For example, the hyperbolic sine function is defined as (ex – e–x)/2 and denoted sinh, pronounced “ shin ”, so that sinh x = (ex – e–x)/2. Definition 1.1 We’ll […]
12.7 Areas Of Surfaces Of Revolution Return To Contents Go To Problems & Solutions 1. Surfaces Of Revolution Consider a sphere of radius r. See Fig. 1.1. We wish to find the surface area of this sphere. For this purpose we set up an xy-coordinate system in such a way that its origin is at […]
16.2.2 Equations With Variable Coefficients – Reduction Of Order Return To Contents Go To Problems & Solutions 1. Second-Order Linear Homogeneous Differential Equations With Variable Coefficients In this section, we’ll use the abbreviations: “DE” for “differential equation”, “GS” for “general solution”, “HDE” for “homogeneous differential equation”, and “HE” for “homogeneous equation”. Recall that a second-order […]
Calculus Of One Real Variable – By Pheng Kim Ving Chapter 6: The Trigonometric Functions And Their Inverses – Section 6.2.2: Differentiation Of The Inverse Trigonometric Functions 6.2.2 Differentiation Of The Inverse Trigonometric Functions Return To Contents Go To Problems & Solutions The Derivatives Of Arcsine And Arccosine Consequently: Similarly: Remark that (d/dx) arccos x […]
16.1.3 First-Order Linear Equations Return To Contents Go To Problems & Solutions 1. First-Order Linear Differential Equations In this section, the abbreviation “DE” stands for “differential equation”. In Section 7.5 Part 3, we encountered the DE dy/dt = ky, or y‘ + by = 0, where b = – k. In this equation, the highest […]
16.3.1 Equations With Constant Coefficients – Undetermined Coefficients Return To Contents Go To Problems & Solutions 1. Second-Order Linear Non-Homogeneous Differential Equations With Constant Coefficients In this section, we’ll use the abbreviations: “DE” for “differential equation”, “GS” for “general solution”, “HDE” for “homogeneous differential equation”, “HE” for “homogeneous equation”, “NHDE” for “non-homogeneous differential equation”, […]
16.2.1 Equations With Constant Coefficients – Characteristic Equation Return To Contents Go To Problems & Solutions 1. Second-Order Linear Homogeneous Differential Equations With Constant Coefficients In this section, we’ll use the abbreviations: “CE” for “characteristic equation”, “DE” for “differential equation”, “GS” for “general solution”, and “PS” for “particular solution”. In the differential equation: y” […]
16.3.2 Equations With Variable Coefficients – Variation Of Parameters Return To Contents Go To Problems & Solutions 1. Second-Order Linear Non-Homogeneous Differential Equations With Variable Coefficients In this section, we’ll use the abbreviations: “DE” for “differential equation”, “GS” for “general solution”, “HDE” for “homogeneous differential equation”, “HE” for “homogeneous equation”, “NHDE” for “non-homogeneous differential […]
10.3 Integration Of Trigonometric Functions Return To Contents Go To Problems & Solutions 1. Basic Trigonometric Integrals Recall from Section 10.1 Part 4 that: For sec x: We group the basic trigonometric integrals together here in the following box. Example 1.1 Evaluate: Solution EOS Go To Problems & Solutions Return To Top Of Page 2. […]
Return To Contents Go To Problems & Solutions Let C be a curve joining points A and B. See Fig. 1.1. Choose the points P0, P1, P2, …, Pn–1, Pn along C such that A = P0 and B = Pn. Denote the length of the chord (line segment) P0 P1 by |P0 P1|, that […]
CALCULUS OF ONE REAL VARIABLE A TUTORIAL By Pheng Kim Ving, BA&Sc, MSc Email: [email protected] Toronto – Canada View If you’re using Internet Explorer and if it doesn’t display the view properly, such as misplaced or (partly-)missing tables or graphics, please try Firefox or Chrome. Welcome To CALCULUS OF ONE REAL VARIABLE!! This website posts […]
Calculus Of One Real Variable – By Pheng Kim Ving Chapter 10: Techniques Of Integration – Section 10.4: Integration Of Powers Of Trigonometric Functions 10.4 Integration Of Powers Of Trigonometric Functions Return To Contents Go To Problems & Solutions An integer has 2 possibilities for parity: even or odd. So a set of 2 integers […]
Calculus Of One Real Variable – By Pheng Kim Ving Chapter 5: Applications Of The Derivative Part 1 – Section 5.3: The First-Derivative Test 5.3 The First-Derivative Test Return To Contents Go To Problems & Solutions 1. Increasing And Decreasing Functions Consider the function f in Fig. 1.1. As x increases, the graph of f […]
11.3 Tests For Convergence Of Improper Integrals Return To Contents Go To Problems & Solutions There are improper integrals that can’t be evaluated by the fundamental theorem of calculus because the antiderivatives of their integrands can’t be found. In this situation, we may still be able to determine whether they converge or not by testing […]
